New estimates for the length of the Erdos-Herzog-Piranian lemniscate
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Let p(z) be a monic polynomial of degree n. Consider the lemniscate L={z:|p(z)|=1}. It has been conjectured that L has the largest length when p(z)=z^n-1. We show that the length of L attains a local maximum at this polynomial and prove the asymptotically sharp bound |L|<2n+o(n).
18 pages
18 pages